This is a tricky question because the two types of claims contradict themselves. On the one hand, disability suggests that you either haven't been able to work in the past 12 months or you can't work for the next 12 months because of a physical or mental impairment. On the other hand, unemployment suggests that you can, in fact, work and that you are searching for a job that meets your skills but cannot find one.
According to Social Security, having unemployment benefits while applying for SS disability will not prevent you from receiving disability benefits. However, a Social Security examiner can use the unemployment filing as evidence in your social security case. He or she may, for example, find that you are applying for jobs for which you deem yourself fit to pursue. If you are, in fact, able to pursue these jobs, the examiner may deny your case because you do not meet the social security disability definition of being "disabled". The examiner may, on the other hand, note that the jobs you are applying for require skills or abilities beyond your capability considering your limitations and experience.
There are obvious risks should you decide to go ahead and apply for Social Security Disability while receiving unemployment benefits. In the state of Texas, you are required to pay back any unemployment benefits that you were not eligible to receive.
Before doing anything, to avoid risk, it may be a wise choice to talk to a Social Security Attorney about your individual case. The Attorney can help you decide whether or not you should try for both or cancel benefits and the best timing for your course of action.